1. Let A and B be C∗ -algebras and J ⊂ A a closed ideal.
(i) Show that if k · kα is a C∗ -norm on A ⊗alg B then there exists a C∗ -norm k · kβ on
(A/J) ⊗alg B such that we have a short exact sequence
0 → J ⊗α B → A ⊗α B → (A/J) ⊗β B → 0.
In particular, (A ⊗alg B) ∩ (J ⊗α B) = J ⊗alg B.
Applying this to the minimal tensor norm, we see that there exists a C∗ -norm k · kβ on
(A/J) ⊗alg B such that we have a short exact sequence
0 → J ⊗ B → A ⊗ B → (A/J) ⊗β B → 0.
(ii) Show that the sequence
0 → J ⊗max B → A ⊗max B → (A/J) ⊗max B → 0
is exact.
(iii) Using (i) and (ii) show that if J and A/J are nuclear then A is nuclear. Show also
that if A is nuclear then J is nuclear.
2. Show that the map x ⊗ y 7→ xt ⊗ y P
on Matn (C) ⊗ Matn (C) has norm ≥ n (in fact, it is
equal to n). Hint: consider the element i,j eij ⊗ eji .
3. For an abelian C∗ -algebra C(X) find f.d. C∗ -algebras Ai and c.c.p. maps φi : C(X) →
Ai and ψi : Ai → C(X) such that ψi ◦ φi → id pointwise.
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